We provide a geometric framework for the construction of non-vacuum black holes whose metrics are stationary and axisymmetric. Under suitable assumptions we show that the Einstein equations reduce to an Einstein-harmonic map type system and analyze the compatibility of the resulting equations. This framework is fundamental to our construction [3] of metric-stationary axisymmetric bifurcations of Kerr solving the Einstein–Klein–Gordon system, and as such, we include specializations of all of our formulas to the case of a time-periodic massive scalar field.

中文翻译：

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固定轴对称黑洞与物质

我们为非真空黑洞的构建提供了一个几何框架，该黑洞的度量是固定的和轴对称的。在适当的假设下，我们证明了爱因斯坦方程简化为一个爱因斯坦谐波映射类型的系统，并分析了所得方程的相容性。这个框架对于我们构造Kerr的公制平稳轴对称分叉[3]求解爱因斯坦-克莱因-哥顿系统是至关重要的，因此，我们将所有公式的专业化包括在时间周期大标量场的情况下。